"America is at her best when she's standing up for the least among us, and the sooner we pass this legislation into law, the better. We are on the right side of history," Graham said Thursday in a statement. "I look forward to leading this long-overdue effort and pushing for a roll-call vote in the Senate."

Graham, a staunch anti-abortion advocate, has long backed the ban. The chance to push for a vote on the issue will no doubt help him appeal to the core Republican voters he's courting in the primaries.

But Graham's proposal highlighted a divide between the interests of the presidential contenders as they claw through the primary season and those of their Senate colleagues looking at tough elections in swing states next year.

Democrats are eager to see Republicans focus on so-called culture war issues, rather than the economy.

They quickly cast Graham's legislation as evidence of the party's "extreme" views and pounced on the handful of swing-state Republican senators, sending out news releases accusing the GOP majority of "attacking women's health."

"In 2016, voters in Ohio — men and women alike — will remember that Rob Portman was more interested in playing politics with women's health than in moving our country forward, strengthening our economy, and working to end the dysfunction in Washington," Sadie Weiner, spokeswoman for the Democratic Senatorial Campaign Committee, declared in one such statement.

Some polling shows that a ban on abortions after 20 weeks of pregnancy isn't an idea that Americans regard as extreme. A 2013 Washington Post poll found 56% of Americans would support such a measure. Other polls have seen similar results.

At the same time, polls have shown a shift to a more socially liberal electorate in recent years. Gallup last month found that 50% of Americans identify as advocates of abortion rights, while 44% of Americans call themselves abortion foes. It was the first time since 2008 that support for abortion rights had a statistically significant advantage, the pollster said.

That's the sort of data that make some Republicans worry about elevating the issue at all in a general election — particularly in swing states like Ohio or New Hampshire, where incumbent Republican Sen. Kelly Ayotte could face a strong challenge.

Time spent on the stump explaining the details of a vote on banning some abortions may play into the Democrats' agenda, they fear.

It's not clear whether GOP senators will face a vote on the ban. Senate Majority Leader Mitch McConnell (R-Ky.), a co-sponsor of the bill, said last year that he would bring it up for a vote, but he has not mentioned timing since the House passed a version of the bill last month.

McConnell spokesman Don Stewart said Thursday he did not have an update.